# Trignometrical Equations Test 1

Total Questions:50 Total Time: 60 Min

Remaining:

## Questions 1 of 50

Question:If $${\sin ^2}\theta = \frac{1}{4},$$then the most general value of $$\theta$$is

$$2n\pi \pm {( - 1)^n}\frac{\pi }{6}$$

$$\frac{{n\pi }}{2} \pm {( - 1)^n}\frac{\pi }{6}$$

$$n\pi \pm \frac{\pi }{6}$$

$$2n\pi \pm \frac{\pi }{6}$$

## Questions 2 of 50

Question:If $${\sec ^2}\theta = \frac{4}{3}$$, then the general value of $$\theta$$is

$$2n\pi \pm \frac{\pi }{6}$$

$$n\pi \pm \frac{\pi }{6}$$

$$2n\pi \pm \frac{\pi }{3}$$

$$n\pi \pm \frac{\pi }{3}$$

## Questions 3 of 50

Question:General solution of the equation $$\cot \theta - \tan \theta = 2$$is

$$n\pi + \frac{\pi }{4}$$

$$\frac{{n\pi }}{2} + \frac{\pi }{8}$$

$$\frac{{n\pi }}{2} \pm \frac{\pi }{8}$$

None of these

## Questions 4 of 50

Question:If $$\cot \theta + \tan \theta = 2{\rm{cosec}}\theta$$, the general value of $$\theta$$ is

$$n\pi \pm \frac{\pi }{3}$$

$$n\pi \pm \frac{\pi }{6}$$

$$2n\pi \pm \frac{\pi }{3}$$

$$2n\pi \pm \frac{\pi }{6}$$

## Questions 5 of 50

Question:If $${\tan ^2}\theta - (1 + \sqrt 3 )\tan \theta + \sqrt 3 = 0$$, then the general value of $$\theta$$ is

$$n\pi + \frac{\pi }{4},n\pi + \frac{\pi }{3}$$

$$n\pi - \frac{\pi }{4},n\pi + \frac{\pi }{3}$$

$$n\pi + \frac{\pi }{4},n\pi - \frac{\pi }{3}$$

$$n\pi - \frac{\pi }{4},n\pi - \frac{\pi }{3}$$

## Questions 6 of 50

Question:The most general value of $$\theta$$satisfying the equations $$\sin \theta = \sin \alpha$$and $$\cos \theta = \cos \alpha$$is

$$2n\pi + \alpha$$

$$2n\pi - \alpha$$

$$n\pi + \alpha$$

$$n\pi - \alpha$$

## Questions 7 of 50

Question:The solution of the equation $$4{\cos ^2}x + 6$$$${\sin ^2}x = 5$$

$$x = n\pi \pm \frac{\pi }{2}$$

$$x = n\pi \pm \frac{\pi }{4}$$

$$x = n\pi \pm \frac{{3\pi }}{2}$$

None of these

## Questions 8 of 50

Question:If $$\sin 3\alpha = 4\sin \alpha \sin (x + \alpha )\sin (x - \alpha ),$$then $$x =$$

$$n\pi \pm \frac{\pi }{6}$$

$$n\pi \pm \frac{\pi }{3}$$

$$n\pi \pm \frac{\pi }{4}$$

$$n\pi \pm \frac{\pi }{2}$$

## Questions 9 of 50

Question:If $$\sin {\rm{ }}\left( {\frac{\pi }{4}\cot \theta } \right) = \cos {\rm{ }}\left( {\frac{\pi }{4}\tan \theta } \right)\,\,,$$ then $$\theta =$$

$$n\pi + \frac{\pi }{4}$$

$$2n\pi \pm \frac{\pi }{4}$$

$$n\pi - \frac{\pi }{4}$$

$$2n\pi \pm \frac{\pi }{6}$$

## Questions 10 of 50

Question:If $$\sin 2\theta = \cos 3\theta$$and $$\theta$$is an acute angle, then $$\sin \theta$$is equal to [

$$\frac{{\sqrt 5 - 1}}{4}$$

$$\frac{{ - \sqrt 5 - 1}}{4}$$

0

None of these

## Questions 11 of 50

Question:If $$4{\sin ^4}x + {\cos ^4}x = 1,$$then x =

$$n\pi$$

$$n\pi \pm {\sin ^{ - 1}}\frac{2}{5}$$

$$n\pi + \frac{\pi }{6}$$

None of these

## Questions 12 of 50

Question:If $$\cos 3x + \sin \left( {2x - \frac{{7\pi }}{6}} \right) = - 2$$, then $$x =$$ (where $$k \in Z$$)

$$\frac{\pi }{3}(6k + 1)$$

$$\frac{\pi }{3}(6k + 1)$$

$$\frac{\pi }{3}(2k + 1)$$

None of these

## Questions 13 of 50

Question:The equation $${\sin ^4}x + {\cos ^4}x + \sin 2x + \alpha = 0$$is solvable for

$$- \frac{1}{2} \le \alpha \le \frac{1}{2}$$

$$- 3 \le \alpha \le 1$$

$$- \frac{3}{2} \le \alpha \le \frac{1}{2}$$

$$- 1 \le \alpha \le 1$$

## Questions 14 of 50

Question:If $$|k|\, = 5$$and $${0^o} \le \theta \le {360^o}$$, then the number of different solutions of 3$$\cos \theta + 4\sin \theta = k$$ is

Zero

Two

One

Infinite

## Questions 15 of 50

Question:If $$0 \le x \le \pi$$and $${81^{{{\sin }^2}x}} + {81^{{{\cos }^2}x}} = 30$$, then x =

$$\pi /6$$

$$\pi /2$$

$$\pi /4$$

$$3\pi /4$$

## Questions 16 of 50

Question:If $$\sin \theta = \sqrt 3 \cos \theta , - \pi < \theta < 0$$, then $$\theta =$$

$$- \frac{{5\pi }}{6}$$

$$- \frac{{4\pi }}{6}$$

$$\frac{{4\pi }}{6}$$

$$\frac{{5\pi }}{6}$$

## Questions 17 of 50

Question:The only value of x for which $${2^{\sin x}} + {2^{\cos x}} > {2^{1 - (1/\sqrt 2 )}}$$ holds, is

$$\frac{{5\pi }}{4}$$

$$\frac{{3\pi }}{4}$$

$$\frac{\pi }{2}$$

All values of x

## Questions 18 of 50

Question:If $$(1 + \tan \theta )(1 + \tan \varphi ) = 2$$, then $$\theta + \varphi$$=

$${30^o}$$

$${45^o}$$

$${60^o}$$

$${75^o}$$

## Questions 19 of 50

Question:Period of $$|2\sin 3\theta + 4\cos 3\theta |$$is

$$\frac{{2\pi }}{3}$$

$$\pi$$

$$\frac{\pi }{2}$$

$$\frac{\pi }{3}$$

## Questions 20 of 50

Question:The period of $${\sin ^4}x + {\cos ^4}x$$is

$$\pi /2$$

$$\pi$$

$$2\pi$$

$$3\pi /2$$

## Questions 21 of 50

Question:If the angles of a triangle $$ABC$$be in A.P., then

$${c^2} = {a^2} + {b^2} - ab$$

$${b^2} = {a^2} + {c^2} - ac$$

$${a^2} = {b^2} + {c^2} - ac$$

$${b^2} = {a^2} + {c^2}$$

## Questions 22 of 50

Question:In triangle $$ABC$$, $$(b + c)\cos A + (c + a)\cos B$$ $$+ (a + b)\cos C =$$

0

1

$$a + b + c$$

$$2(a + b + c)$$

## Questions 23 of 50

Question:$$\angle A + \angle C = {90^o}$$In$$\Delta ABC,$$ if $$2(bc\cos A + ca\cos B + ab\cos C) =$$

0

$$a + b + c$$

$${a^2} + {b^2} + {c^2}$$

None of these

## Questions 24 of 50

Question:In $$\Delta ABC,$$ $${\rm{cosec }}A(\sin B\cos C + \cos B\sin C) =$$

$$c/a$$

$$a/c$$

1

$$c/ab$$

## Questions 25 of 50

Question:In $$\Delta ABC$$, $$c\cos (A - \alpha ) + a\cos (C + \alpha ) =$$

$$a\cos \alpha$$

$$b\cos \alpha$$

$$c\cos \alpha$$

$$2b\cos \alpha$$

## Questions 26 of 50

Question:In $$\Delta ABC$$, $$\frac{{\cos A}}{a} + \frac{{\cos B}}{b} + \frac{{\cos C}}{c} =$$

$$\frac{{{a^2} + {b^2} + {c^2}}}{{abc}}$$

$$\frac{{{a^2} + {b^2} + {c^2}}}{{2abc}}$$

$$\frac{{2({a^2} + {b^2} + {c^2})}}{{abc}}$$

$${a^2} + {b^2} + {c^2}$$

## Questions 27 of 50

Question:If the angles $$A,B,C$$of a triangle are in A.P. and the sides $$a,b,c$$ opposite to these angles are in G. P. then $${a^2},{b^2},{c^2}$$ are in

A. P.

H. P.

G. P.

None of these

## Questions 28 of 50

Question:If the sides of a triangle are p,q and$$\sqrt {{p^2} + pq + {q^2}}$$, then the biggest angle is

$$\pi /2$$

$$2\pi /3$$

$$5\pi /4$$

$$7\pi /4$$

$$5\pi /3$$

## Questions 29 of 50

Question:The perimeter of a $$\Delta ABC$$is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is

$$\frac{\pi }{6}$$

$$\frac{\pi }{3}$$

$$\frac{\pi }{2}$$

$$\pi$$

## Questions 30 of 50

Question:Point D, E are taken on the side BC of a triangle $$ABC$$such that $$BD = DE = EC$$.If $$\angle BAD = x$$, $$\angle DAE = y$$, $$\angle EAC = z$$, then the value of $$\frac{{\sin (x + y)\sin (y + z)}}{{\sin x\sin z}} =$$

1

2

4

None of these

## Questions 31 of 50

Question:If in a triangle ABC side $$a = (\sqrt 3 + 1)$$cms and $$\angle B = {30^o},$$ $$\angle C = {45^o}$$, then the area of the triangle is

$$\frac{{\sqrt 3 + 1}}{3}c{m^2}$$

$$\frac{{\sqrt 3 + 1}}{2}c{m^2}$$

$$\frac{{\sqrt 3 + 1}}{{2\sqrt 2 }}c{m^2}$$

$$\frac{{\sqrt 3 + 1}}{{3\sqrt 2 }}c{m^2}$$

## Questions 32 of 50

Question:If in a right angled triangle the hypotenuse is four times as long as the perpendicular drawn to it from opposite vertex, then one of its acute angle is

$${15^o}$$

$${30^o}$$

$${45^o}$$

None of these

## Questions 33 of 50

Question:In a$$\Delta ABC,$$if $$b = 20,c = 21$$and $$\sin A = 3/5$$, then $$a =$$

12

13

14

15

## Questions 34 of 50

Question:Let D be the middle point of the side BC of a triangle ABC. If the triangle ADC is equilateral, then $${a^2}:{b^2}:{c^2}$$is equal to

$$1:4:3$$

$$4:1:3$$

$$4:3:1$$

$$3:4:1$$

## Questions 35 of 50

Question:If the sides of triangle be 6, 10 and 14 then the triangle is

Obtuse angled

Acute angled

Right angled

Equilateral

## Questions 36 of 50

Question:In any $$\Delta ABC$$if $$a\cos B = b\cos A$$, then the triangle is

Equilateral triangle

Isosceles triangle

Scalene

Right angled

## Questions 37 of 50

Question:If A is the area and 2s the sum of 3 sides of triangle, then

$$A \le \frac{{{s^2}}}{{3\sqrt 3 }}$$

$$A \le \frac{{{s^2}}}{2}$$

$$A > \frac{{{s^2}}}{{\sqrt 3 }}$$

None of these

## Questions 38 of 50

Question:If in a triangle $$ABC$$right angled at $$B,s - a = 3$$, $$s - c = 2$$, then the values of a and c are respectively

2, 3

3, 4

4, 3

6, 8

## Questions 39 of 50

Question:The area of triangle $$ABC,$$ in which $$a = 1,\;b = 2$$, $$\angle C = 60^\circ$$is

$$\frac{1}{2}$$

$$\sqrt 3$$

$$\frac{{\sqrt 3 }}{2}$$

$$\frac{3}{2}$$

## Questions 40 of 50

Question:In a triangle $$ABC$$, if $$b + c = 2a$$ and $$\angle A = 60^\circ ,$$ then $$\Delta ABC$$ is

Scalene

Equilateral

Isosecles

Right angled

## Questions 41 of 50

Question:In any triangle ABC, $$a\cot A + b\cot B + c\cot C =$$

$$r + R$$

$$r - R$$

$$2(r + R)$$

$$2(r - R)$$

## Questions 42 of 50

Question:If the radius of the circumcircle of an isosceles triangle $$PQR$$ is equal to $$PQ( = PR),$$then the angle P is

$$\frac{\pi }{6}$$

$$\frac{\pi }{3}$$

$$\frac{\pi }{2}$$

$$\frac{{2\pi }}{3}$$

## Questions 43 of 50

Question:The angle of elevation of a tower at a point distant d meters from its base is$${\rm{3}}0^\circ$$. If the tower is 20 meters high, then the value of d is

$$10\sqrt 3 m$$

$$\frac{{20}}{{\sqrt 3 }}m$$

$$20\sqrt 3 m$$

$$10$$m

## Questions 44 of 50

Question:The angle of elevation of the top of the tower observed from each of the three points $$A,B,C$$on the ground, forming a triangle is the same angle $$\alpha$$. If R is the circum-radius of the triangle ABC, then the height of the tower is

$$R\sin \alpha$$

$$R\cos \alpha$$

$$R\cot \alpha$$

$$R\tan \alpha$$

## Questions 45 of 50

Question:From a point a metre above a lake the angle of elevation of a cloud is a and the angle of depression of its reflection is $$\beta$$. The height of the cloud is

$$\frac{{a\sin \,(\alpha + \beta )}}{{\sin \,(\alpha - \beta )}}$$

metre$$\frac{{a\sin \,(\alpha + \beta )}}{{\sin \,(\beta - \alpha )}}$$ metre

$$\frac{{a\sin \,(\beta - \alpha )}}{{\sin \,(\alpha + \beta )}}$$ Metre

None of these

## Questions 46 of 50

Question:If the angle of depression of a point A on the ground from the top of a tower be $$30^\circ$$, then the angle of elevation of the top of the tower from the point A will be

$$60^\circ$$

$$45^\circ$$

$$30^\circ$$

None of these

## Questions 47 of 50

Question:The angle of depression of a ship from the top of a tower 30 metre high is $${60^0}$$, then the distance of ship from the base of tower is [MP PET 1988; Pb. CET 2003]

30 m

$$30\,\sqrt 3 \,\,m$$

$$10\sqrt 3 \,m$$

10 m

## Questions 48 of 50

Question:The angle of elevation of a stationary cloud from a point 2500 m above a lake is $${15^0}$$ and the angle of depression of its reflection in the lake is $${45^0}$$. The height of cloud above the lake level is

$$2500\,\sqrt 3 \,metres$$

2500 metres

$$500\,\sqrt 3 \,metres$$

None of these

## Questions 49 of 50

Question:A person is standing on a tower of height $$15(\sqrt 3 + 1)\,m$$ and observing a car coming towards the tower. He observed that angle of depression changes from $${30^0}$$ to $${45^0}$$ in 3 sec. What is the speed of the car

36 km/hr

72 km/hr

18 km/hr

30 km/hr

## Questions 50 of 50

Question:Two men are on the opposite side of a tower. They measure the angles of elevation of the top of the tower $${45^0}$$ and $${30^0}$$ respectively. If the height of the tower is 40 m, find the distance between the men

$$40\sqrt 3 \,m$$